Wavelet analysis of a signal transforms the signal into a time-scale domain. The wavelet domain may produce different interpretations of a signal than other common transforms like the Fourier Transform or the Short-Time Fourier Transform. Wavelets have been used in applications for data analysis, data compression, and audio and video coding. Wavelets may be described using filter bank theory. A signal may be passed through a series of complementary low-pass and high-pass filters followed by decimators in a two-channel, analysis filter bank. At each stage in the filter bank, the input signal is broken down into two components: a low-pass, or coarse, part and a high-pass, or detailed, part. These two components are complimentary as a result of the way the filters are created. A Wavelet Transform further decomposes the coarse part from each iteration of the filter bank. A Wavelet Packet Transform provides the option of decomposing each branch of each stage. Many interesting decomposition structures may be formed using Wavelet packets.
Wavelet theory may also be described using Linear Algebra Theory. In the discrete-time case, an input signal can be described as an N-dimensional vector. If the input signal is infinite, the input vector is infinite. If the input signal is finite and is n samples long, the vector is n-dimensional. An N-dimensional vector lies in the Euclidean N vector sub-space. A signal transform, such as a Fourier Transform or a Wavelet Transform, projects the input vector onto a different sub-space. The basis vectors of the new sub-space also form a basis for the original sub-space, N.
A Wavelet Transform includes two main elements: a high-pass filter followed by decimation, and a low-pass filter followed by decimation. These two operations can be thought of as two separate transforms. The high-pass channel projects the input vector onto a high-pass sub-space, and the low-pass channel projects the input vector onto a low-pass sub-space. The high-pass sub-space may be called the Wavelet Sub-Space, W, and the low-pass sub-space may be called the Scaling Sub-Space, V. The low-pass channel in the filter bank can be iterated numerous times, creating many levels in the transform. With each iteration, the input vector is projected onto another Wavelet and Scaling Sub-Space. The Wavelet Sub-Space at level j may be labeled as Wj, and the Scaling Sub-Space at level j may be labeled as Vj.
Wavelet Packet Transforms allow any channel to be iterated further. With each iteration, the input vector is being projected onto another Wavelet and Scaling Sub-Space. A Wavelet Packet Tree decomposition requires a slightly different naming convention for the various sub-spaces. A node in the tree structure may be described by its depth i and position j. The Wavelet Sub-Space at depth i and position j may be labeled as Wi,j, and the Scaling Sub-Space may be labeled as Vi,j. It may also be noted that for the Wavelet Packet Transform, the Wavelet Sub-Space may only be located at odd j positions (with numbering beginning at zero) and the Scaling Sub-Space may only be located at even j positions.